The Born Rule as a Geometric Measure on Projective State Space—And an Octonionic Outlook

Presented is a geometric reformulation of the Born rule for finite-dimensional quantum systems. The state space is identified with complex projective space equipped with its canonical Fubini–Study geometry. Three structural axioms — locality in projective distance, invariance under projective unitaries, and additivity/non-contextuality for orthogonal decompositions — are shown to reduce the problem of transition probabilities to the framework of Gleason's theorem, thereby uniquely determining the transition probability P ([ψ] → [ϕ]) = |⟨ϕ|ψ⟩|². The argument provides a transparent geometric reformulation and interpretation of Gleason’s theorem. I then show that any relativistic Dirac spinor theory automatically realizes this geometry locally, so the Born rule is inherited without further assumption. Finally, I formulate a precise conjecture for an octonionic analogue involving the exceptional group G₂ (or F₄/E₈), then illustrate the idea with a simple finite toy model derived from the seven imaginary octonions and the Fano plane.